Fuligni 1997, pp. 155-156 deals with the Republic of Minerva. Michael J. Oliver, American citizen born in Lithuania, fond of numismatics, decided once to found a new nation. He got in touch with British authorities to put up a tax-free state in the Turks and Caicos Islands, and of course failed with this. He found in an atlas that there were two atolls in formation at the north of the territory of Tonga, known as Minerva since a ship named this way had wrecked because of them. Oliver and a friend, Morris Bud Davis, bought a ship in May 1972, engaged some men, and with this brought some sand on Minerva, to make it become a real island. When there was enough sand to make it possible to walk on Minerva, they proclaimed the Republic, Davis becoming president, and Olivier making money with his book (A new constitution for a new country) and the coins (with goddess Minerva) he made for his state.

Tuphou IV of Tonga was rather unhappy to see this republic be born in his
kingdom, and sent the Tongan army there, where the soldiers took the flags of the new republic [sic] and replaced them with the Tongan flag.

Reported flag of Minerva: golden torch on blue. I received from George Cruikshank a very good image of the Minerva flag. I slightly corrected
it according to an image in Vexilologie no. 6. Note that Balough 23 report a change of flag. Blue with a narrow red stripe from the lower hoist to the upper fly. In the blue fly is centred a yellow five-pointed star.
Jaume Ollé, 1-3 May 2001

I think that I read about this in The People's Almanac, II by Irving Wallace, Amy Wallace, and David Wallechinsky. (...) As I recall, the latitude and longitude of Minerva Reef were given, and I found the spot well to the south of Tongan territorial waters. The proponents of this project asked the King of Tonga whether he had any claim to the reef, and he did not, so they staked their own claim to it, and were utterly bewildered when the king sent his armed forces to seize it for Tonga.John S. Ayer, 9 June 2001

From The People's Almanac #2 by David Wallechinsky and Irving Wallace,
1978:

Established in 1972 by a declaration of sovereignty by a group of Californians, the Republic of Minerva has more claim to authenticity than most micronations because it actually has some land, although it disappears at high tide. The republic consists of two coral reefs 17 miles apart in the South Pacific Ocean some 3,400 miles southwest of Honolulu and 915 miles northeast of Auckland, New Zealand. President Morris C. Bud Davis, a former project engineer with North American Rockwell, runs this domain from the living room of his suburban house in Justin, California. Originally the plan was to attract a population of 60,000 to a fancy sea resort, called Sea City, which was to be constructed on the reefs, where residents would have "no taxation, welfare, subsidies, or any form of economic interventionism". Since the main income was to be from the registration of international cargo ships, the Sea City project was dropped as impractical for lack of funds. A ship was, however, purchased to carry sand out to the homeland for "a major landfill project". Minerva also had a political skirmish with the Kingdom of Tonga over ownership of the reefs that nearly led to
a war. When Davis made his original claim by building a small stone tower on one reef, with a flasher on it and the Minervan flag (a torch on a blue field), King Taufa'ahau Tupou IV set sail from Tonga 200 miles to the northeast aboard the ship Olovaba with the 100-man strong Tonga Defense Force (recruited from among the prisoners). They tore down the flag and read a Tongan proclamation of sovereignty. Said Davis: "We can't for the life of us understand why the king should suddenly decide he wants the reefs". While Minerva remains uninhabited, the conflict is dormant.

The reef in question is probably the same as Falcon Island, which pops up every 10 years, does its stuff, and then goes down again. That island falls under Tongan jurisdiction; as soon as it is up, the king orders the Tongan flag to be planted there. Sometimes it's too late...Jarig Bakker, 9 June 2001

On the website for "The Principality of Minerva", a so-called "microstate",
there is a picture of their flag. See
http://www.minervanet.org. Jim Brown, 11 August 2007

See especially
http://www.minervanet.org/factsheet.htm.
The link between the previous republic and this principality is explained here:
http://www.minervanet.org/foreign/history.htm:
"In 2003 the Principality was established as the new and official government of
the Minervan state, with no official ties to any representatives of the old
Republic regime; it is noteworthy to mention that while the Principality
was not directly endorsed by the founders of the Republic and that the
individuals who were associated with the Republic and its subsequent collapse no
longer wish to be associated with the Minervan State in any way, they did
communicate to the Principality that that they had no continuing interest in
maintaining the Republic and that the Principality had full right to establish
itself as the legitimate government of the isles, becoming the obvious and
logical successor."
Hence this totally different micronational flag...Olivier Touzeau , 11 August 2007